On Second Order Diff. Operators with Different B. C.& Weight Functions: Mathematics Ordinary Differential Equations

The aim of this book is to present some types of spectral problems, which includes second order differential equation with different boundary conditions for each type. The book studies the properties of eigenvalues and estimation of normalized eigenfunctions with different cases of weight functions, in both regular and irregular cases. It also estimates the Green?s functions to the spectral problem. Firstly, the spectrum to model, which is defined in chapter two, are real and the normalized eigenfunctions to that model with different cases of weight functions are estimated. Secondly, it is shown that the eigenvalues are located in upper half plane and are complex (i.e. pure imaginary); the asymptotic behavior of eigenvalues is shown through the model approaches to the infinity and the normalized eigenfunctions are proved to be uniformly bounded in both regular and irregular cases, whenever the weight functions are smooth and ?(x)?C^2 (0,a) and q(x)?C(0,a). Finally, we found the upper bound for norm to the Green?s function which is defined in chapter four in the regular case.